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We study a model of public decision-making in simple public goods economies with moral hazards and adverse selection. Economic agents must invest resources (or provide effort) to discover their own preferences. We consider direct revelation mechanisms based on sampling. A sample of agents is drawn in the population, and each member of the sample reports a preferences type to a Principal. The determinants of the "representative sample" size are studied. The structure and magnitude of effort and sampling costs affects the optimal number of representatives. If the net social value of the effort is high, first and second best optimality require a maximal sample (or "direct democracy"). If, on the contrary, effort is too costly, the recourse to samples ("representative democracy") is justified as a second best. To obtain the results, we not only take effort and revelation incentives into account, but also restrict decision rules to satisfy an additional property of robustness to opportunistic manipulation by the Principal, which forbids the use of a priori knowledge in public decision procedures.
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DISCUSSION PAPER SERIES
ABCD
www.cepr.org
Available online at:
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No. 6417
ON THE OPTIMAL NUMBER
OF REPRESENTATIVES
Emmanuelle Auriol and Robert J. Gary-Bobo
DEVELOPMENT ECONOMICS and
PUBLIC POLICY
ISSN 0265-8003
ON THE OPTIMAL NUMBER
OF REPRESENTATIVES
Emmanuelle Auriol, IDEI, Toulouse School of Economics and ARQADE
Robert J. Gary-Bobo, Université Paris 1 Panthéon-Sorbonne, Paris School of
Economics and CEPR
Discussion Paper No. 6417
August 2007
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Copyright: Emmanuelle Auriol and Robert J. Gary-Bobo
CEPR Discussion Paper No. 6417
August 2007
ABSTRACT
On the Optimal Number of Representatives*
We propose a normative theory of the number of representatives based on a
stylized model of a representative democracy. We derive a simple formula, a
"square-root theory" which gives the number of representatives in parliament
as proportional to the square root of total population. Simple econometric tests
of the formula on a sample of a 100 countries yield surprisingly good results.
These results provide a benchmark for a discussion of the appropriateness of
the number of representatives in some countries. It seems that the United
States have too few representatives, while France and Italy have too many.
The excess number of representatives matters: it is positively correlated with
indicators of red tape, barriers to entrepreneurship and perceived corruption.
JEL Classification: D7, H11 and H40
Keywords: constitution design, incentives, number of representatives and
representative democracy
Emmanuelle Auriol
IDEI
University of Toulouse I
32 rue de Mounic
09340 Verniolle
FRANCE
Email: eauriol@cict.fr
For further Discussion Papers by this author see:
www.cepr.org/pubs/new-dps/dplist.asp?authorid=127618
Robert J. Gary-Bobo
Université Paris 1
Panthéon-Sorbonne
Maison des Sciences Economiques
106-112 Boulevard de l'Hôpital
75647 Paris Cedex 13
FRANCE
Email: garybobo@univ-paris1.fr
For further Discussion Papers by this author see:
www.cepr.org/pubs/new-dps/dplist.asp?authorid=125284
* We thank Nabil Al Najjar, David Austen-Smith, Tim Besley, Helge Berger,
Jacques Drèze, Gabrielle Demange, Jean-Pierre Florens, Françoise Forges,
Roger Guesnerie, Arye Hillman, Mamoru Kaneko, Jean-Jacques Laffont,
Jean-François Laslier, John Ledyard, Philippe Mongin and Thomas Palfrey for
their help and comments, as well as seminar audiences at the Free University
of Berlin, UC Dublin, Universities of Tilburg, Toulouse, Stockholm, University
of Wisconsin, Madison, University of Illinois at Urbana-Champaign,
Northwestern University, Boston College, Caltech, UCLA, and CEPR (Public
Policy Program) for useful remarks. The present paper is a deeply revised and
extended version of a manuscript circulated under the same title in the year
2000 (DP no 1286 of the Center for Mathematical Studies, in Economics and
Management Sciences, Northwestern University).
Submitted 25 July 2007
1 Introduction
The production of public goods affects the well-being of large number of citizens, whereas a
typically much smaller number of individuals is in charge of public decisions. This is true at
almost all levels of society: there are parliaments at the national level, councils at the local
levels and even committees within public or private organizations. The presence of costs
associated with the acquisition of information and with the preparation of decisions plays
a major role in this concentration of power. The forces driving the division of labor help
understanding the emergence of representatives. As a counterpart, protection against the
opportunistic behavior of these representatives becomes a major justification of collective de-
cision rules. This paper studies the trade-off between the need to economize on decision costs,
suggesting that a small number of individuals should specialize in public decision-making,
and the democratic requirement that decisions should reflect the citizens’ true preferences.
We focus on a theory of the optimal number of representatives, that we also test with the
help of political data.
We adopt a two-stage approach to constitutional design,
1
with a constitutional and
a legislative stage, to derive the optimal number of representatives. In contrast to most of
the recent work on constitution design, we completely black-box elections and voting and
construct what could be called a reduced-form theory of representative democracy. The leg-
islators’ assembly is modeled as a random sample of preferences, drawn from the population
of citizens. The randomly chosen representatives do not vote; they use a nonmanipulable,
revealing mechanism instead. This mechanism reveals the representatives’ preferences and
efficient public-decisions are carried out by a self-interested executive. During the prelimi-
nary constitutional stage, fictitious Founding Fathers choose decision rules behind the veil
of ignorance, so as to maximize the expected total sum of citizens’ utility. The Founding
Fathers know that no agent is benevolent. It follows from this that the executive’s hands
must be tied as much as possible and that representatives must be provided with incentives
to reveal preferences truthfully. In addition, our Founding Fathers know that they don’t
1
On this question, see the survey in Mueller (2003), and the discussion of some recent contributions below.
2
know the distribution of preferences that will prevail in society. The novelty of this article
is that we do not assume that this distribution is common knowledge. A robust mechanism
is therefore required, in the following particular sense: among nonmanipulable mechanisms,
the Founding Fathers pick a decision rule that maximizes expected utility against a vague (or
noninformative) prior relative to citizens’ preferences. Using a well-known technique from
Bayesian statistics, a limiting argument is used to derive the effect of the Founding Fathers’
ignorance on the optimal mechanism.
2
Robustness in this sense can be understood as a political stability requirement. The
Founding Fathers know that society is going to evolve, but they cannot anticipate in which
way. A constitution could not last for more than 200 years if it was tailored too closely to a
particular preference profile. It must work under very different distributions of preferences.
Our model singles out a well-defined robust mechanism, that happens to be a sampling
Groves mechanism. Statistical sampling properties then yield an optimal sample size, which
trades off the direct and opportunity costs of representatives with the welfare loss induced
by representation (i.e., the loss due to the fact that a subset of citizens make decisions).
A “square-root formula” for the optimal number of representatives directly follows
from this stylized model of representation. The rule is then tested with the help of a sample
of more than 100 countries, and we find that our square-root theory is almost true and
reasonably robust. World data is well-approximated by a number of national representatives
proportional to N
0.4
, where N is the country’s total population. We also identify the US,
France and Italy as outliers. The former lie below the regression line; the latter two much
above. The same model does not fit the data on the 50 US State Legislatures very well. We
conclude that the historical rigidity of political US institutions is likely to be the cause of this
lower quality of fit. Indeed constitutional History shows that the representation ratio has
constantly decreased for more than 200 years in the United States. Tocqueville (1835, part I,
Chap. VIII, p 190, footnote) already noted the fact that the representation ratio decreased
from 1 representative for every 30,000 inhabitants in 1792, to 1 over 48,000 in 1832. This
trend has not been reversed ever since, the ratio reaching a record low of 1 over 611,000
2
The most technical aspects of our theory are presented in Auriol and Gary-Bobo (2007).
3
in the recent years. Furthermore, the number of seats in the House of Representatives has
reached a ceiling of 435 in 1910.
3
According to our results, the US Lower and Upper Houses
should have a total of 807 members.
We finally check for correlation of the number of representatives with some indices
measuring openness to trade, the costs of setting up a new firm (i.e. “red tape”), the degree
of state interference in markets, and perceived corruption.
4
The results are clearly that the
number of representatives matters: it is positively and significantly correlated with state
interference, red tape, and corruption. More precisely, we cannot reject the fact that it is the
excess number of representatives (i.e., the actual number less the number predicted by the
N
0.4
formula) which in fact matters for red tape and the degree of state interference.
The question of the appropriate number of seats in US Parliament has been posed a
long time ago by the founding fathers and opponents of the American Constitution. James
Madison addressed the question in a famous passage of Federalist n
10:
In the first place, it is to be remarked that however small the Republic may be,
the Representatives must be raised to a certain number, in order to guard against
the cabals of a few; and however large it may be, they must be divided to certain
number, in order to guard against the confusion of a multitude.
Madison, Federalist 10 (in Pole (1987), p 155).
The Anti-Federalist writers have emphasized a related point:
The very term, representative, implies, that the person or body chosen for this
purpose, should resemble those who appoint them (...). Those who are placed in-
stead of the people, should possess their sentiments and feelings, and be governed
by their interests, or, in other words, should bear the strongest resemblance of
those in whose room they are substituted. (...) Sixty-five men cannot be found in
3
This number has been fixed by statute in 1929. See O’Connor ans Sabato (1993), p 191.
4
We use indices constructed by Barro and Lee (1994), Djankov et al. (2002), Treisman (2000), and
Transparency International, respectively.
4
the Unites States, who hold the sentiments, possess the feelings, or are acquainted
with the wants and interests of this vast country.
Essays of Brutus, III, 1787 (in Storing (1981), p 123)
Some essential ideas are condensed in the above quotations. In the paper we interpret
them as stating that there exists a tradeoff between the need to protect citizens against the
dictatorship of a minority and that of reducing the costs of public decision-making. As
far as we know the problem of the optimal number of legislators has been studied by a
handful of economists only.
5
In contemporary writings, Buchanan and Tullock (1962) are
clearly the forerunners of the approach followed here. Thinking about constitutional design,
they developed a theory of the optimal constitution based on 4 variables: rules for choosing
representatives; rules for deciding issues in assemblies; the degree of representation (i.e., the
proportion of total population elected); and the basis of representation (i.e., for instance,
the geographical basis). Buchanan and Tullock’s approach is clearly normative, insofar as
the goal of the analysis is to fix the 4 variables in order to minimize the expected sum of
decision-making and external costs of institutions. Another forerunner is Stigler (1976), who
sketched a theory of the degree of representation and proposed some regression work on the
number of representatives in relation to total population in the US States.
A small (but influential) number of authors belonging to the Public Choice school
has played with the ideas emphasized here a long time ago: following Dahl (1970), Mueller
et al. (1972) discuss random representation. Tullock (1977) went as far as to ponder over
the practical possibility of using pivotal mechanisms in the US Congress to make public
decisions. In the present paper, our intention is not to advocate the recourse to random choice
of legislators, or Groves mechanisms in practice, but to propose a model of representative
democracy in reduced form and to derive a formula for the optimal number of representatives.
We are not the first to adopt a “reduced-form approach” to models politics. For
instance, in Becker (1983), political parties and voting receive little attention because “they
5
This problem is essentially distinct from that of fair representation or apportionment, that has been
studied quite extensively, e.g. Balinski and Young (2001). Our theory is not related to L. S. Penrose’s
(1946) square-root formula. Penrose’s formula determines the size of a country’s delegation in supra-national
institutions like the UN or EU, not the number of representatives itself.
5
are assumed mainly to transmit the pressure of active groups”. Becker (1983) defines political
equilibrium as a Nash equilibrium among pressure groups using expenditures on influence
as strategic variables. Such an approach has both limitations and advantages. More recent
contributions in which a common agency model is used to study public policy-making can
also be viewed as employing a reduced-form methodology (see e.g. Dixit et al. (1997)).
There has been a recent revival of interest in the normative method among writers
in Political Economy, Voting Theory and Mechanism Design. Our normative approach does
not rely on the existence of a benevolent planner and our self-interested executives are
clearly in line with the citizens-candidate approach of Osborne and Slivinski (1996) and
Besley and Coate (1998). The two-stage approach to Constitutional Design recently received
further impetus from Aghion and Bolton (2003), Barbera and Jackson (2004), Erlenmaier
and Gersbach (2001). Some contributions explore voting rules, or alternative collective
decision procedures, with the idea of improving efficiency through a better expression of the
intensity of preferences (e.g., Casella 2005). An extension of optimal taxation theory to a
dynamic setting in which citizens must rely on a non-benevolent politician to implement
redistribution policies has been proposed by Acemoglu et al. (2007). Political Economy
considerations are now more and more introduced in normative analysis and give rise to new
constraints, in addition to revelation or incentive constraints. Our underlying philosophy
has much in common with that of these recent contributions.
In the following, Section 2 presents our basic assumptions; Section 3 develops our
model of representation; Section 4 derives the robust representation mechanism and the
square-root theory of the optimal number of representatives. Section 5 presents the empirical
results: econometric tests of the square-root theory in the world and among the US State
legislatures; it also discusses the empirical relevance of the number of representatives by
showing its impact on red tape, state interference and corruption indices. A few technical
results are proved in the appendix.
6
2 The Model
2.1 Basic Assumptions
We consider an economy composed of N + 1 agents, indexed by i = 0, 1, ..., N. A public
decision, denoted q, must be chosen in a set Q. Agent i will pay a tax denoted t
i
. This tax
must be interpreted as a subsidy if it is negative. Each agent’s utility depends on the public
decision and the tax.
Assumption 1. (Quasi-linearity) Utilities are quasi-linear, and defined as v
i
(q) t
i
, where
v
i
, is a private valuation function.
These valuation functions can be viewed as random draws in some unknown probability
distribution P on a set of admissible valuation functions V .
Assumption 2. (Statistical Independence) For all i, the v
i
are independent drawings from
the same distribution P on V . The distribution P has a well-defined mean.
Society comprises three types of individuals. Agent i = 0, called the executive, is in charge of
executing the collective decision q. After some relabelling if necessary, agents i = 1, ..., n are
representatives; and agents i = n + 1, ..., N are passive citizens. The task of representatives
is to transmit information on preferences. We assume the following.
Assumption 3. (Cost of Representation) Each representative pays a fixed cost F , i.e., if i
is a representative, then i’s utility is v
i
(q) t
i
F .
This cost can be viewed as the sum of direct and opportunity costs of becoming a represen-
tative or, alternatively, as an elementary form of information-acquisition cost paid by agent
i to obtain information about one’s own preferences v
i
. Under the former interpretation,
citizens use resources to transmit information to the collective decision system. Under the
latter interpretation, individuals do not know their own utility function and must incur costs
to become aware of their own preferences. The two interpretations are compatible.
A representation is basically a random sample of n N agents (or, equivalently, a
random sample of preferences v = (v
1
, ..., v
n
)).
7
Assumption 4. (Perfect Representation) The n representatives are independent random
drawings in the probability distribution P .
In practice, it is doubtful that voting mechanisms would produce an unbiased random sample
of preferences. On the one hand, Assumption 4 might seem a rather naive idealization, but
can be defended if our goal is to construct a normative theory of representative democracy
and to determine the optimal number of representatives. On the other hand, the idea
of unbiased random representation provides a desirable simplification, putting the entire
electoral process in a black box. Representation by lot existed in some societies of the
past (see Hansen (1991), Manin (1997));
6
it has been discussed by political scientists (Dahl
(1990)) and is still used to select juries in some countries. However, the representation biases
induced by voting systems cannot be studied with the simplest form of this model. We will
nevertheless continue to work with this convenient idealization.
Definition 1 (Representation Mechanism). A representation mechanism is an array of
functions (f, t), where f is a collective decision rule mapping representatives’ reports about
preferences bv = (bv
1
, ..., bv
n
) into Q, i.e., q = f(bv), and a list of tax functions denoted t =
(t
0
, t
1
,...t
N
), satisfying the budget constraint
P
N
i=0
t
i
= 0.
By definition, the constitution specifies (f, t) for every possible value of n, but n itself is not
fixed in the constitution.
2.2 The First-Best Optimum
We can now compute the first-best optimum in the above defined economy. The standard
Utilitarian, first-best Bayesian decision maximizes the function
EW = E
P
(
N
X
i=0
(v
i
(q) t
i
) | (bv
1
, ..., bv
n
)
)
nF, (1)
6
The ancient greeks, in Athens, used random drawings to choose their legislators. The Athenian people’s
assembly itself, with its 6000 members, was in fact a random sample of the citizen population. Each citizen
attending a session of this Assembly would receive the equivalent of a worker’s daily wage. Socrates was
sentenced to death by a jury of 501 randomly drawn citizens (see Hansen (1991)).
8
with respect to q in Q, subject to the budget constraint
P
N
i=0
t
i
= 0, where E
P
denotes the
expectation with respect to probability P . Given that individual preferences are independent
draws in probability distribution P , this is equivalent to solving the problem:
max
qQ
(
(N + 1 n)E
P
(v(q)) +
n
X
i=1
bv
i
(q) nF
)
, (2)
where E
P
(v(.)) is the average utility function in the population. To understand how this
first-best optimum looks like, assume for example that preferences are quadratic, with a
single-dimensional parameter θ, i.e., v
i
(q) = θ
i
q q
2
/2 and that q is a nonnegative real
number. Assume in addition that P is such that E(θ) = µ and V ar(θ) = σ
2
. With these
specifications, (2) becomes
max
qQ
(
q
"
n
X
i=1
b
θ
i
+ (N + 1 n)µ
#
(N + 1)
q
2
2
nF
)
. (3)
This immediately yields the optimal decision
q
= f
(
b
θ
1
, ...,
b
θ
n
) =
1
N + 1
n
X
i=1
b
θ
i
+ (N + 1 n)µ
!
; (4)
Substituting (4) into EW , taking the expectation with respect to the distribution of θ
i
,
yields the ex ante expected welfare associated with the optimal decision rule f
. After some
easy computations, we obtain
EW (f
) =
2
2(N + 1)
+
(N + 1)µ
2
2
nF, (5)
where we make use of the fact that the
b
θ
i
are i.i.d. This function being linear with respect
to n, we can state the following result.
Proposition 1. With quadratic preferences, the first-best optimum has two possible values:
either n
= N + 1, if σ
2
> 2(N + 1)F , (i.e., a Direct Democracy), or n
= 0, if σ
2
2(N + 1)F , (i.e., a ”Reign of Tradition”).
The interpretation of Proposition 1 is easy. If the dispersion of preferences is large
enough with respect to costs of representation, then direct democracy is first-best optimal.
9
In other words, if the individual cost of participating in the collective decision process F
is small, or if the number of citizens is small, then democracy must be direct. The only
other case is not a democratic constitution: we call this “Reign of Tradition” because it
is not dictatorship (which would correspond to n = 1). In the Reign of Tradition, no
citizen is endowed with the power of deciding on behalf of others and we can view the
public decision as being the result of “Tradition”, i.e., f
= µ. Another equivalent view
is that the decision is made by a disembodied benevolent planner. This arrangement is
optimal only if the dispersion of preferences is small or if the population is large and if in
addition, the prior mean of preference parameters µ is common knowledge. Proposition 1 is
disappointing, because it never prescribes a representative democracy, in which the solution
would be interior, i.e., 0 < n
< N + 1. The most likely case is one in which F is small
but nonnegligible, N is very large, and tastes do not differ in an extreme way, which seems
to indicate that the Reign of Tradition would always be the recommended solution. This
failure to pick a representative democracy as a solution is not essentially due to the fact
that expected welfare is linear with respect to n under quadratic preferences (and to the
fact that total representation costs nF are linear). It stems from the assumption that the
distribution of preferences is common knowledge. Indeed, if this is the case, if in addition
N is large and if the dispersion of tastes is reasonable, by the Law of Large Numbers, µ
is an excellent estimator of the true population-mean of individual valuations and it is not
useful to ask citizens about their taste parameters. Our claim is that there is something
wrong with the above definition of the optimum, because the model describes a world in
which information is not really decentralized. The model is that of an abstract benevolent
planner, endowed with prior knowledge of the distribution of preferences (i.e., (µ, σ) in the
quadratic example), but in a large economy with quadratic preferences, knowing µ means
knowing almost everything that is useful: Democracy is useless.
In Section 4 below, we propose a different model in which information is fully decen-
tralized, the distribution of tastes is not common knowledge and democratic representation
is a useful (and only) way of producing information. Section 3 will first provide some basic
definitions and pose the representatives’ incentive compatibility problem.
10
3 Representation and Incentives
To give formal content to the idea of an impartial and benevolent point of view on society,
we assume the existence of fictitious agents called the Founding Fathers (hereafter the FF).
The FF are in charge of writing the constitution; they are assumed benevolent, Bayesian,
and Utilitarian, and they do nothing in the economy, apart from setting constitutional rules.
These FF know that, once the set of rules embodied in the constitution will be applied,
there will not exist a single omniscient, impartial and benevolent individual to carry out
public decisions. A disembodied “social planner” is not assumed to play an active role. This
imposes restrictions on the set of admissible mechanisms, described in sub-section 3.1. The
ensuing preference revelation problem is studied in sub-section 3.2.
3.1 Basic Constitutional Principles
The FF apply some important principles. First, Separation of Power holds: the executive
cannot be a representative. Second, a Subsidiarity Principle applies. According to Definition
1 above, a representation mechanism is an array of functions (f, t). To work in practice, such
a mechanism needs to be fully specified and this specification may depend on a number of
controls or parameters. We need to allocate the power to choose the exact value of these
parameters, and these choices may open some possibilities of manipulation. This motivates
the following definition.
Definition 2 (Subsidiarity Principle). With the exception of the number of representatives
n itself, if the parameters needed to fully pin down and implement mechanism (f, t) are not
specified in the constitution and are not provided for by the representatives according to
constitutional rules, then they are chosen by the executive.
The Subsidiarity Principle simply says that the executive will fill all the gaps in the public
decision process. It can of course be dangerous to let the executive choose crucial parameters
freely, because this executive is endowed with unknown preferences (v
0
is a random draw in
P ) and would be tempted to pursue private goals.
11
Third, the FF also apply a principle of “Anonymity” (or “Equality” in a weak sense),
which imposes equal treatment of indistinguishable individuals. This forces equal tax treat-
ment of all passive citizens, because their preferences are unknown (and there is no basis for
discrimination among them). Let t
0
denote the tax of agents i = n + 1, ..., N and i = 0. The
budget constraint can thus be rewritten as follows:
n
X
i=1
t
i
+ (N + 1 n)t
0
= 0. (6)
3.2 Incentive Compatibility
The decision rule f, as well as taxes t, should be immune to manipulations of the rep-
resentatives and of the executive. Appealing to the Revelation Principle, we require the
representation mechanism (f, t) to be direct and revealing. But the agents’ beliefs about
others’ preferences are not common knowledge and are unknown to the FF. Mechanism
(f, t) must therefore be revealing whatever the beliefs of the representatives. In this con-
text, it almost immediately follows that (f, t) must be revealing in dominant strategies (see
Ledyard (1978)), i.e., for all i = 1, ...n, for all v
i
, bv
i
, and v
i
,we must have
v
i
(f(v)) t
i
(v) v
i
(f(bv
i
, v
i
)) t
i
(bv
i
, v
i
),
where, as usual, we denote v
i
= (v
1
, ..., v
i1
, v
i+1
, ..., v
n
) and v = (v
i
, v
i
).
Because of the subsidiarity principle, the self-interested executive could choose the free
parameters of (f, t) to maximise his (her) own utility v
0
. These parameters must therefore
be fixed in the constitution. In our simple model, revelation in dominant strategies plus
”mast-tying” of the executive, put together, define non-manipulability.
Definition 3 (Non-Manipulability). A representation mechanism (f, t) is nonmanipulable if
it is revealing in dominant strategies and if all its parameters are specified in the constitution.
This definition means that, in addition to the revelation property, there are no free pa-
rameters that the executive could manipulate. It is possible to prove (see the appendix,
for comments and a formal statement), that under separation-of-powers, subsidiarity and
12
anonymity principles, non-manipulable mechanisms must assume the following form: the
decision rule f(.) must maximize an objective which is the sum of an arbitrary function k
and of the utilities reported by representatives, i.e.,
f(bv) arg max
qQ
(
k(q) +
n
X
i=1
bv
i
(q)
)
. (7)
And for all i = 1, ..., n, representatives must be subjected to the following tranfer schedules:
t
i
(bv) =
X
j6=i
bv
i
(f(bv)) k(f(bv)) + m(bv
i
), (8)
where m is an arbitrary fixed function that does not depend on bv
i
. Finally, arbitrary functions
k, and m must be fixed in the constitution. Obviously, the choice of these crucial parameters
cannot be left to the executive, because the choice of k can distort decisions radically, while
the choice of m can distort transfers. We assume that the FF are constrained to choose
f(.) in this set of nonmanipulable mechanisms. When k 0, the class of nonmanipulable
mechanisms boils down to the well-known class of Clarke-Groves mechanisms, but restricted
to a random subset of agents called the representatives.
7
Note that these mechanisms are budget-balanced by construction, because there is
at least one citizen which is not a representative (i.e., at least agent 0 does not report
about his (her) preferences). In other words, passive citizens form a sink used to finance the
revelation incentives of the representatives. It follows that there are no inefficiencies due to
budget imbalance (budget surplus), as in the usual theory of pivotal mechanisms. The only
welfare losses are due to the fact that the information on preferences used by a representation
mechanism is not exhaustive; in other words, social costs are caused by sampling errors.
4 Robust Representation Mechanisms under Decen-
tralized Knowledge
The novelty of our approach is that we have assumed that the FF do not know the prob-
ability distribution of citizens’ preferences P , and they know that nobody knows it. We
7
On Groves mechanisms, see Clarke (1971), Groves (1973), Green and Laffont (1979), Holmstrom (1979),
Moulin (1986). On sampling Groves mechanisms, see Green and Laffont (1977), Gary-Bobo and Jaaidane
(2000).
13
add the constraint of decentralized knowledge to the assumptions of asymmetric information
and individual opportunism: the probability distribution of preferences P is not common
knowledge.
The fact that the FF do not know P poses a problem because they cannot fully specify
the expected (or average) welfare function that they would like to maximize by means of
the choice of a constitution. There are several ways of modeling behavior under ignorance
in decision theory. One is to use a non-probabilistic representation and a maximin principle
or, some more sophisticated variant in which the decison-maker uses a set of probability
distributions. The constitution would then be chosen so as to maximize welfare against the
worst-case scenario. Another approach is to choose decision rules that are optimal against a
non-informative, or vague prior. In contrast, this is a purely Bayesian approach. We choose
this latter route here. There is a mathematical difficulty in the representation of a decision
maker’s complete prior ignorance because a uniform distribution on the real line (or on the
set of integers) doesn’t exist.
8
It follows that a situation of complete prior ignorance can be
approached by limiting arguments, letting the prior’s variance go to infinity.
4.1 The Founding Fathers’ Objective
We first assume that the FF constrain themselves to choose a decision rule that satisfies
“Weak Utilitarianism”.
9
Definition 4 (Weak Utilitarianism). The decision rule f should maximize the expected
utility E
P
0
(v(q)) with respect to q in Q for some probability distribution P
0
on V .
Imposing Weak Utilitarianism in the sense of Definition 4 means that the decision rule must
maximize some weighted sum of utilities. Given that the FF are assumed to be Utilitarians,
this requirement is very weak, because P
0
can be chosen arbitrarily.
We can now derive what we call robust mechanisms. It is easy to see that, under
non-manipulability, the FF’s goal is essentially to choose the arbitrary function k. If the set
8
Bayesian statisticians have developed the theory of improper priors. See e.g. Bernardo and Smith (1994).
9
But the utilitarian principle could also be derived, in the manner of Harsanyi (1955), by assuming that
the FF are rational decision-makers, and choose the objective function behind the veil of ignorance.
14
of possible utilities V is convex, the weak utilitarianism requirement imposes to choose k of
the form k(q) = bv
0
(q), where b 0 is a scalar and v
0
is in V , for otherwise, the maximand
k(q)+
P
n
i=1
bv
i
(q) could not be expressed as the expected social utility for some probability
distribution.
10
Formally, the social surplus function is defined as
W (f) = nF +
N
X
i=0
v
i
(f). (9)
The FF would like to maximize the expected value of this social welfare with respect to
decision rule f(.), subject to nonmanipulability and weak utilitarianism. In this perspective,
we assume that they have a “prior on priors”, i.e., a distribution B on possible priors P ;
and we assume that B is uninformative this represents the FF’s lack of knowledge about
the true distribution of citizens’ preferences. Expected social welfare can be expressed as
E
B
E
P
(W ), were W is defined by (9).
4.2 The Founding Fathers’ Beliefs
The only problem is now to give formal content to the idea that the FF will choose a nonma-
nipulable f (.) so as to maximize E
B
E
P
(W ) under a vague (or non-informative) probability
B. Such a decision rule will simply be called robust. Intuitively, this can be done by a sim-
ple limiting argument, if P belongs to a family with a finite vector of parameters, by letting
the precision of B converge towards zero (or equivalently, by letting the variance-covariance
matrix of B go to infinity). This definition is involved, but the intuition is simple: find the
nonmanipulable mechanism which maximizes expected welfare under the “veil of ignorance”,
using a non-informative prior.
Auriol and Gary-Bobo (2002), (2007) have studied the existence of robust mechanisms
in this sense, assuming that the set of public decisions is finite, that individual preferences
profiles can be any vector and that these vectors are multivariate normal (i.e., P is multivari-
ate normal, according to the Founding Fathers’ beliefs). Thus, the domain of preferences is
general, but a normality assumption is used. As in portfolio theory, we can weaken the nor-
10
In other words, bv
0
(q)+
P
n
i=1
bv
i
(q) is proportional to a weighted average of utilities for all b and v
0
.
15
mality requirement, but will obtain a tractable model only if utility is assumed quadratic.
We follow this direction here, because our theory can easily be illustrated in the classic
quadratic-preference setting.
Assumption 5. (Quadratic preferences) Decision q is a real number, and,
V =
v(q) = θq
q
2
2
, θ R
. (10)
In this simple setting, the true probability distribution P is just a one-dimensional distribu-
tion of the taste parameter θ, with a finite mean µ
P
, and a finite variance σ
2
P
. In this case,
we also assume that the FF do not know (µ
P
, σ
2
P
), but that they are endowed with a prior
B on possible pairs (µ
P
, σ
2
P
). In addition we assume the following:
E
B
(µ
P
) = bµ, E
B
(σ
2
P
) = bσ
2
, and V ar
B
(µ
P
) = bz
2
, (11)
where bµ, bσ
2
, bz
2
are themselves finite, and where bµ is the mean of the possible means, bσ
2
is
the mean of the possible variances, and bz
2
is the variance of the possible means. The prior
variance of θ, from the FF’s point of view, is denoted V ar
F F
(θ), and admits the well-known
decomposition,
V ar
F F
(θ) = V ar
B
[E(θ|P )] + E
B
[V ar(θ|P )]
= bz
2
+ bσ
2
.
We propose the following simple formal definition.
Definition 5 (Robust Representation Mechanism). A mechanism (f, t) is robust if it is the
limit of a sequence (f
k
, t
k
) of mechanisms, such that each (f
k
, t
k
) maximizes E
B
k
(E
P
W ) on
the set of nonmanipulable mechanisms, where (B
k
) is a sequence of priors with the property
that that bz
2
k
goes to +, while bσ
2
k
/bz
2
k
goes to zero.
To understand this definition, assume that all possible P distributions have the same variance
σ
2
P
= bσ
2
, but that their mean µ
P
is unknown to the FF. To approach complete ignorance,
we let the variance of the possible means, i.e. bz
2
, go to infinity. As indicated above, a more
general definition is of course possible, but would be more technical.
16
4.3 Derivation of the Robust Mechanism in the Case of Quadratic
Utility
Under Assumption 5, nonmanipulability and weak utilitarianism force us to choose a utility
function of the form v
0
(q) = αq q
2
/2 with a weight β 0 and a decision rule f
∗∗
(.), such
that
f
∗∗
(
b
θ
1
, ...,
b
θ
n
) arg max
q
(
q
n
X
i=1
b
θ
i
nq
2
2
+ β
αq
q
2
2
)
, (12)
assuming that each representative i reports
b
θ
i
. We immediately find
f
∗∗
(
b
θ
1
, ...,
b
θ
n
) =
P
n
i=1
b
θ
i
+ αβ
n + β
. (13)
Let now W
P
(α, β) be the expected welfare for a given distribution P and f
∗∗
as above. We
have
W
P
(α, β) = E
P
(
f
∗∗
(
b
θ)
N
X
i=0
θ
i
(N + 1)f
∗∗
(
b
θ)
2
2
)
nF. (14)
We then compute the expected value of W
P
with respect to the FF’s prior B. Some compu-
tations yield the following formula.
Lemma 1.
E
B
[W
P
(α, β)] =
n + β
N + 1
2
nbσ
2
(n + β)
2
+
b
2
(N + 1)
2(n + β)
2
(2αbµ α
2
)
+
n(N + 1)
(n + β)
2
n
2
+ β
(bµ
2
+ bz
2
) nF. (15)
For proof, see the appendix.
For given B, the best mechanism is obtained as a maximum of W = E
B
[W
P
(α, β)] with
respect to (α, β). We find the following result.
Lemma 2. For given B, the optimal values of α and β are α
∗∗
= bµ, and
β
∗∗
=
(N + 1 n)bσ
2
bσ
2
+ (N + 1)bz
2
. (16)
For proof, see the appendix.
17
This solution can be rewritten as a function of the ratio ζ = bσ
2
/bz
2
. We immediately find
the limit of β
∗∗
as ζ 0,
lim
ζ0
β
∗∗
= lim
ζ0
(N + 1 n)ζ
ζ + (N + 1)
= 0.
Under decentralized knowledge, the only robust mechanism entails v
0
(q) = bµq q
2
/2 and
β
∗∗
= 0 and therefore, the arbitrary function k must be set identically equal to 0. This
mechanism is a sampling Groves mechanism. To make a public decision, it relies on the
representatives’ reports only. Formally, we have just proved the following result.
Proposition 2. Under Assumptions 1-5, the only robust mechanism f
∗∗
(bv) maximizes
P
n
i=1
bv
i
(q), with transfers t given by (8) above.
Since preferences are assumed quadratic, we get q
∗∗
= f
∗∗
(bv
1
, ..., bv
n
) = (1/n)
P
n
i=1
b
θ
i
. In
fact, the same sampling Groves mechanism is robust in our sense with a much more general
set of preferences, but at the cost of some normality assumption (on P , not on B).
11
The sampling Groves mechanism solves a number of difficult problems of a representa-
tive democracy simultaneously. It saves on the costs of producing information on preferences,
captured by the fixed cost F , because of sampling; it ensures honest revelation of their pref-
erences by representatives in a very strong sense (i.e., Groves mechanisms are revealing in
dominant strategies); and finally, once subjected to the incentive transfer system (8) (see
also Proposition A1 in the appendix), every representative adheres to the same social ob-
jective (i.e., every representative agrees with the objective of maximizing
P
n
i=1
bv
i
(q)). The
interpretation of this result is that the legislative bargaining process yields an approximate
Pareto optimum, insofar as the representation is a correct mirror image of the population’s
preferences. Of course, this nice solution is obtained for a somewhat simplified economy with
quasi-linear preferences (i.e., a public good economy with possibilities of compensation).
Remark that, if we let the prior’s variance bz
2
go to zero instead, while bσ
2
remains
bounded, then, we find lim
ζ→∞
β
∗∗
= N + 1 n . This means that the FF know the
distribution of preferences in society for sure. In this case, the recommended solution is the
11
Again, see Auriol and Gary-Bobo (2007).
18
standard Bayesian mechanism of sub-section 2.2, where v
0
(q) = bµq q
2
/2 = E
P
(v(q)) and
N + 1 n is the appropriate weight of v
0
in the expected welfare function E[W | q, bv
1
, ..., bv
n
]
(and N + 1 n is also the number of passive citizens). In this latter case, the sampled
agents represent only themselves, while in the robust mechanism, sampled agents are truly
representatives: they stand for the entire society. This is a major difference. We now show
that in this setting, an optimal number n
∗∗
can be interior, i.e., 0 < n
∗∗
< N + 1, in sharp
contrast with the standard Bayesian first-best analysis presented in sub-section 2.2.
4.4 Optimal Number of Representatives
We can now compute the optimal number of representatives, denoted n
∗∗
. Substituting the
robust decision rule f
∗∗
(θ) = (1/n)
P
n
i=1
θ
i
in the expression for expected welfare yields
W =
N + 1
2
(bµ
2
+ bz
2
) +
bσ
2
2
1
n
1
N + 1
(N + 1)bσ
2
2
nF. (17)
Define q
N+1
=
1
N+1
P
N
i=0
θ
i
. If we compute the first-best surplus in an economy with N + 1
agents, using complete knowledge of the preference profile and then take expectations, we
find
E
B
E
P
"
q
N+1
N
X
i=0
θ
i
(N + 1)
q
2
N+1
2
#
nF = (N + 1)E
B
E
P
q
2
N+1
2
nF
=
bσ
2
2
+
N + 1
2
(bµ
2
+ bz
2
) nF. (18)
Let q
n
=
1
n
P
n
i=1
θ
i
. Under the robust mechanism, we get the following expression of welfare,
W = (N + 1)E
B
E
P
q
n
q
N+1
q
2
n
2
nF. (19)
Taking the difference of expressions (18) and (19), we find the welfare loss (with respect to
the complete information first-best),
L(n) =
(N + 1)
2
E
B
E
P
(q
N+1
q
n
)
2
. (20)
It is then easy to check that
L(n) =
1
n
1
N + 1
(N + 1)bσ
2
2
; (21)
19
and it follows that expression (17) is first-best surplus, minus the cost of representatives,
minus the welfare loss due to the fact that some information on preferences is not reported.
The optimal number of representatives n
∗∗
trades off the cost of an additional representative
with the benefit of reducing the welfare loss, i.e., n
∗∗
minimizes nF + L(n). The repre-
sentatives protect citizens against arbitrary public decisions, but there is a social cost of
representation.
Observe that the social cost of representation nF + L(n) does not depend on bz
2
(which can thus be arbitrarily large). It follows that if the FF had prior information on the
variance of preferences bσ
2
, they could compute the optimal number of representatives under
the robust mechanism. At the time of the writing of the constitution, the FF may have had
some knowledge of F , N and bσ, but have been well aware that these parameters vary with
time. The constitution should therefore allow for changes in the optimal n. In other words,
the number of seats in parliament should not be fixed by the constitution.
12
The first-order condition for a maximum of W with respect to n, viewed as a real
number, is easy to compute and yields F + (1 + (N + 1)/2n
2
))bσ
2
= 0. From this we derive
the following result.
Proposition 3. With quadratic preferences, the optimal number of representatives is 1 plus
the integer part of
n = bσ
r
N + 1
2F
. (22)
If n is smaller than 1, we choose n
∗∗
= 1. This appears when F is very large, or bσ very
small. In this case, a single person (a “technocrat”) will make the public decision.
13
If, on
the contrary, F is small, or bσ is very large, we get n
∗∗
= N (everybody is a representative,
12
This does not mean that it should be free. In our stylized model, the rule to change the number of seats
could be fixed by the constitution, while the number itself is not. In practice, it is usually possible to change
the number of representatives without amending the constitution. For instance, in France the number of
representatives is determined by an ”organic act” which is stronger than ordinary law but weaker than the
constitution.
13
But the technocrat is not a dictator, because, when bσ is small, preferences tend to be very close, and
there is a consensus about the optimal decision.
20
except the executive), and we obtain a direct democracy. In this latter case, the first-best is
almost implemented.
14
Proposition 3’s formula suggests an econometric model of the form:
log(n) = log(bσ) + (1/2) log(N + 1) (1/2) log(2F ) + , (23)
where is a zero-mean, random error term. This formulation is simple and natural. The
three factors determining the number of representatives are: the variance of preferences,
the size of the population, and the costs of representation. This simple model fits the data
remarkably well, as we now show.
5 Empirical Assessment, on Political Data
To empirically predict the size of representative political institutions, we have assembled a
data set for a sample of 111 countries that possess a parliament or representative assemblies.
The total number of representatives, n, is expressed in numbers of individuals. It includes
all representatives at the national (or federal) level, e.g., the sum of the members of the
lower and upper houses, when a country has a bicameral legislature. We do not count
the representatives in local governments, in the member states of a federation, or in the
district or city-councils. Our point of view has been to study the determinants of national
representation sizes. The population size, denoted N in the following, is expressed in millions
of citizens. These two pieces of information were extracted from The Europa World Year
Book (1995). To fix ideas, the USA are in the sample with n = 535 and N = 260.341.
France has 898 representatives (d´eput´es plus enateurs).
15
We have estimated the same
model separately with data relative to the 50 parliaments of the US States.
5.1 The Square-Root Model with World Data
To get a preliminary view of the empirical relevance of the theory, we have first regressed
the total number of representatives n (expressed in numbers of individuals) on population
14
In the first best, strictly speaking, we have n
= N + 1 (see sub-section 2.2).
15
In the case of the UK, adding some 1221 peers to 651 MPs (in 1995) would have created an outlier: so
we decided not to add the Peers.
21
size N (expressed in millions of citizens). A first regression of the form n = a + bN yields
significant estimates of a and b, but with a poor R
2
(the adjusted R
2
is .27). By contrast,
a much better adjustment is obtained when, as suggested by theory, log(n) is regressed on
log(N) plus a constant (without any constraint). We find the following result,
log(n) = 4.324
(75.26)
+ 0.41
(17.63)
log(N) (24)
In the above regression, t-statistics are between brackets. The adjusted R
2
is .74, and the
global F-statistic is a highly significant 311.23. Moreover, the estimated constant, 4.324, and
the estimated coefficient, 0.41, are both relatively close to the theoretical predictions which
are 6.561 = (1/2)(log(10
6
)/2 log(2)), and 0.5 respectively. In particular, the estimated
power of N is below 1/2, but not much. The estimated constant captures some of the effect
of the omitted variables. But the result is surprisingly good for such a crude regression. See
Fig.1, for a plot of n against N in the studied sample.
Insert Figure 1 here.
According to the theory, a more heterogeneous population should lead to larger par-
liaments, and countries where the cost of representation is high should have smaller ones.
It is difficult to capture population heterogeneity σ
2
and the per capita (opportunity) cost
of representation F in the regression. We can only hope to find proxies for σ and F . We
were not able to find a database, or even international comparison studies on the social cost
to maintain a representative assembly. We have checked some national accounts in order
to get a sense of the costs involved. They are quite large. For instance, in the US, the
legislative branch funding rose from USD 2.8 billions in 2001 to USD 4.3 billions, requested
in 2007 (a 57% growth). The average annual cost of maintaining one representative can
hence be estimated in 2006 around USD 8 millions, or 210 times the US GNP per capita.
In Australia, the cost of maintaining the elected representatives in federal parliament was
estimated at AD 400 millions in 2004. This puts the average annual cost of maintaining
one representative around AD 2 millions (i.e., USD 2.6 millions), more than 100 times the
Australian GNP per capita. In Canada, the total cost was CD 468 millions in 2004-2005.
22
The average annual cost per representative is then CD 5.5 millions (i.e., USD 4.95 millions),
more than 200 times the Canadian GNP per capita. None of these amounts include the
costs of running an election (i.e., campaigning and administrative costs). It is obvious that
there is some variance in the unit cost of representation: in GDP per capita terms, US and
Canadian representatives cost twice as much as Australian representatives.
16
According to
the theory, this should play a role in determining the number of representatives, given that F
is in fact the sum of opportunity and direct costs per representative. To capture the impact
of these costs on the size of the legislative bodies, we rely on several proxies. We add the
logarithm of the GDP per capita to the regression. We also add the logarithm of the total
national tax revenue, expressed as a percentage of GDP (denoted T AXREV ). The idea
is that wealthier countries and wealthier governments will not find it difficult to maintain
a large assembly. The expected sign of the the tax-revenue variable is thus positive. The
expected sign of the per capita GNP is ambiguous: if it acts as a proxy for the opportunity
costs of representation, the coefficient might as well be negative. We also add the logarithm
of the average government wages (denoted GOV W AGE), expressed as a percentage of GDP.
This variable provides an indication on the representatives’ wages, and that of their staffs.
It is related to the per capita cost of maintaining the assembly. The expected sign of this
wage variable is thus negative. Unfortunately, we have the wage data for 62 countries only.
We first add the 3 variables sequentially to the log(n) regression. The results are presented
in Table 1 below.
Column (1) is just the crude regression presented above. The quality of fit increases sub-
stantially with additional controls, as indicated by the adjusted R
2
of columns (2)-(5), which
is around 83%. The coefficients of the GNP per capita and of the tax revenue are positive;
the coefficient of the government wages is negative, as predicted by the theory. We next run
a regression with the 3 variables simultaneously, reported in column (2) of Table 1. There
are only 62 countries because of the missing wage data. We next run a regression without
the GNP variable because it is not significant in the regression above and one without the
16
This is presumably due to the fact that, contrary to their US and Canadian counterparts, Australian
representatives do not set their own wages and benefits (they are fixed by an independent court).
23
wage variable, because it reduces our sample by half. Coefficients are fairly robust, and
particularly the coefficient on log(N): its values are closer to the theoretical prediction in
columns (2) and (5), with a value of .44, possibly because the simplest log-linear model has
an omitted-variable bias.
TABLE 1. Dependent Variable: log
n
(1) (2) (3) (4) (5)
Constant 4.32 2.98 5.46 4.28 3.98
(75.26)*** (7.9)*** (27.16)*** (10.0)*** (6.55)***
log(N) 0.41 0.44 0.4 0.41 0.44
(17.63)*** (14.71)*** (16.12)*** (16.88)*** (12.9)***
log(GNP) 0.04 0.04 0.02
(1.21) (1.53) (0.55)
log(TAXREV) 0.34 0.17 0.24
(2.98)*** (1.96)* (1.76)*
log(GOVWAGE) -0.12 -0.18
(-2.08)** (-1.82)*
DENSITY -0.0001 -0.0001 - 8.5×10
5
(-2.48)** (-2.96)*** (2.26)**
GINI -2.53 -1.79 -1.1
(-5.48)*** (-3.79)*** (-1.73)*
ELF -1.74 -1.25 -1.37
(-3.85)*** (-3.07)*** (-2.41)**
GINI×ELF 3.54 2.69 2.95
(3.49)*** (3.06)*** (2.43)**
No. Obs. 111 62 93 93 55
R
2
0.74 0.83 0.83 0.85 0.86
Adjusted R
2
0.74 0.82 0.82 0.84 0.83
Sum squared Resid 14.5 4.25 8.89 7.74 3.26
Columns (1)–(5) were estimated by ordinary least squares. White heteroskedastic-consistent standard errors
are used to calculate t-statistics, which are reported in parentheses. Significance is denoted by *** (1%); **
(5%); * (10%).
In order to measure population heterogeneity σ, we have tried different variables. We
first added population density (inhabitants per square kilometer, divided by 10,000) in the
basic regression. The intuition is that people who leave far apart do not interact much,
and may differ more. They have different ways of life, different jobs, face different climates,
etc. High density countries (e.g., Japan) should present less heterogeneity than low density
countries (e.g., the US). According to the theory, the sign of the density coefficient should
then be negative. The data, which give the number of inhabitants per square kilometer in
24
1996, are from the World Bank (World Development Indicator 1998). As predicted by the
theory, the sign of the density coefficient is negative and significant, but it is quite small in
absolute value (to fix ideas, a decrease of the density variable by 1 unit in the US would
result in one additional representative only).
It also seems reasonable to assume that countries including many different linguistic
and ethnic groups are more heterogeneous. We thus add the ethno-linguistic fractionalization
index, denoted ELF , as an explanatory variable.
17
This index varies between 0 and 1 and
gives the probability that two randomly chosen individuals do not speak the same language.
Higher ethno-linguistic fractionalization indices may signal more heterogeneous populations
and the sign of the ELF coefficient should then be positive. This index is known for 93 of
the 111 countries considered.
Another variable that might reflect population heterogeneity is the Gini coefficient
of each country. Gini coefficients provide a measure of heterogeneity in the sense that a
large coefficient signals an unequal distribution of income in the population. If we admit
that more unequal societies are more heterogeneous, everything else being equal, they should
have larger representatives assemblies. The coefficient of the Gini variable should then be
positive. For the sake of comparison with the ELF coefficient, the data, which are from
the World Bank (World Development Indicators, 1998), are divided by 100 to be normalized
between 0 and 1. When added separately to the log(n) equation, the ELF and Gini index
coefficients are both significant and negative, which is the “wrong” sign, as shown by columns
(3)-(5) in Table 1. This seemingly contradicts the theoretical result that more population
heterogeneity in the population should be associated with a larger representation. However,
when we consider the joint effect of wealth inequality and ethno-linguistic fragmentation
(i.e., the product ELF × Gini), the coefficient of this interaction variable is both significant
and positive, which is the expected sign. The net effects of Gini and ELF are negative, but
the effect of income inequality dampens as the level of ethno-linguistic fragmentation rises.
Symmetrically, the effect of higher ethno-linguistic fragmentation is mitigated as the level of
income inequality rises.
17
Source: Easterly and Levine, World Bank, 2003
25
It is not clear that the variables used as regressors are really good proxies for the
degree of preference heterogeneity in a given country. It might well be that Gini and ELF
poorly capture the relevant aspects of heterogeneity. Clearly, other variables should be
included in the regression to better capture the effect of preference heterogeneity on the size
of representations. From the statistical point of view, Gini and ELF are also potentially
endogenous variables, at least in the long run, but an instrumentation of these variables is
outside the scope of the present paper. Another possibility is of course that countries with
more unequal income distributions (and higher ELF indices) are characterized by a form of
power capture by the richest (and (or) by some ethnic groups). Some of the political regimes
considered in the sample are far from being ideal democracies.
Finally, we have added several other variables, capturing some aspect of countries’
history and legal institutions, as robustness checks.
18
The only additional variables that
turned out to be significant are dummies, denoted DEM46 and OCDE indicating when the
country experienced 46 years of continuous democracy and a member of OCDE, respectively
(see Table A1, in the Appendix). DEM46 has a negative coefficient: everything else equal,
old democracies have less representatives than young democracies. This suggests that the
number of seats in legislatures is characterized by institutional rigidity. OCDE has a positive
coefficient and seems to capture the same effects as T AXREV and GNP .
To sum up, results indicate that the number of representatives n is not determined
by a constant sampling rate: it increases less than proportionately with the size N of the
population, since according to our estimates on the 111 countries, n (0.3)N
0.4
(when N
is the count of inhabitants). This formula yields 475 representatives for a 100 millions of
people. This finding has been shown to be robust, and supports fairly well (it is indeed a
close variant of) the square-root theory of the optimal number of representatives derived
above.
Apart from assessing whether the normative principles underlying the analysis are at
18
P ERCEN T P ROT is the percentage of protestants in the country; OECD indicates an OECD country,
TRANS a former socialist (i.e;, transition) country; F EDERAL a country with a federal structure (e.g.,
the US, Germany); AF RICA indicates a country form the African continent; F ORM BRIT COL indicates
a former British colony and the UK; COMLAW indicates a country with a Common Law system; DEM 46
equals 1 if the country has been democratic in all 46 years (1950-1995) (see Treisman (2000)).
26
work in actual political institutions, our empirical exercise can also be viewed as a first step
to compare the political systems of different countries, in terms of representation size. Table
A2, in the appendix, gives the list of countries, the observed value of n and its fitted value bn,
computed with the help of Regression 4 in Table 1. For the missing data the simple regression
1 in Table 1 is used. These figures are reported in italic. Figure 2 plots the actual number
n of representatives (denoted REP RE) against the predicted number of representatives bn
(denoted REP REF ). This plot has been drawn with the results of Regression 4 in Table 1.
Insert Fig. 2 here
We find that France and Italy have “too many” representatives, whereas the United
States do not have “enough” of them (i.e., they lie below the regression line). In fact,
both France and Italy have more representatives than the US. According to our results,
the US Lower and Upper House should have 807 members instead of 535. The political
US institutions seem characterized by historical rigidity. We confirm this result in the next
sub-section.
5.2 Number of Representatives in the US State Legislatures
Using the data provided by McCormick and Turner (2001), for US state legislatures in 1996,
we have tested the square-root model with the 50 US States, adding the state senators and
representatives together to form the n variable. The state population (in million) is for 2005.
We find that the crude log-linear regression yields
log(n) = 4.696
(52.35)
+ 0.172
(3.32)
log(N), (25)
the adjusted R
2
is equal to .21, and the global F = 14.16, with exactly 49 observations
(t-statistics are in parentheses). Adjustment quality is mediocre as compared to the results
obtained with world data. Among the US states, New Hampshire, Nebraska and Nevada are
outliers. New Hampshire has a plethoric n = 400 representatives and we have removed this
state from the sample. If we take the 50 States, the simple log-linear regression above yields
a coefficient of 0.14 on log(N), with a t of 2.55 , and R
2
= .12. Fig. 3, which includes New
27
Hampshire, shows that the quality of the adjustment is bad. Removing another outlier will
not change results much (this will not increase the coefficient on log(N) substantially).
Insert Fig.3 here
We then added the population density in 2000 and representatives’ salary, averaged for 1995-
97 (known for 40 states from McCormick and Turner 2001). Without New Hampshire, this
yields the following regression, which is a little closer to our square-root model:
log(n) = 4.723
(41.13)
+ 0.218
(3.09)
log(N) + 5.11
(2.47)
(10
4
)Density 0.218
(2.03)
(10
6
)Salary. (26)
The adjusted R
2
= .35, the global F = 4.78 (significant at 3%) and t-statistics are in
parentheses. We already observed that the US are an outlier among nations in the world. A
strong dependence on history, and a very slow historic speed of adjustment of the number
of representatives seem to be the main explanations for the low quality of the adjustment.
Population has increased enormously in some US states during the 20th century, without
much change in the number of state representatives. The number of federal representatives in
Washington D.C. has itself been fixed by statute in 1929 (see O’Connor and Sabato (1993)).
The US seem to be characterized by an extreme form of rigidity in these matters.
These results and those obtained with world data suggest that some countries have
an excessive number of representatives while others have too few. It seems important to
analyze the impact of having too few or too many representatives on the performance of
institutions. With too few representatives, public decisions could well be biased in favor of
active minorities, to the detriment of under-represented (or less organized) groups. Casual
observations also suggest that the corruption level could be higher in countries characterized
by an “excessive” number of representatives. We indeed find a result of that sort below.
5.3 The Number of Representatives and Red Tape
We now examine the link between the number of representatives and barriers to business
entry, entrepreneurship and trade. The Public Choice school offers a theory relating lobby
activity and the number of representatives (see Mueller (2003)): the influence of each rep-
resentative should decrease with their number; lobbies would be ready to pay more to buy
28
a vote when the number of seats in parliament is low. Becker’s (1983) approach is also
compatible with the existence of an impact of the number of representatives, if n affects the
pressure groups’ influence functions through changes in the “pressure technology”.
To check for the presence of a possible influence of the number of representatives on
variables related to lobbying activity, we consider in turn 3 indices: (i) a measure of trade
openness, denoted F REEOP (and taken from Barro and Lee (1994)); (ii) a measure of
the direct cost of meeting government requirements to open a new business, expressed as
a fraction of 1999 GDP per capita, denoted SUNKCOST (due to Djankov et al. (2002));
(iii) and a measure of whether state interference hinders business development, denoted
ST AT EINT ERF (due to Treisman (2000)). We regress the three variables on log(n),
log(N), log(GNP ), and many controls. The only significant variables are LAND, which
measures the country’s land surface in millions of square kilometers, DEM46 and T RANS
dummies (defined above). The regressions, presented in Tables 2 and 3, check for correlation,
not necessarily for causality. Yet, as explained above, the number of representatives is largely
predetermined and rigid in most countries: it is either fixed by the constitution, or by statutes
with a high rank in the hierarchy of norms, that cannot be changed easily. It is of course
endogeneous in the long run. Hence, the number of representatives has good chances of being
”exogenous” compared to F REEOP , SUNKCOST and ST AT EINT ERF , that can be
changed easily. Table 2 and 3 presents the results obtained with OLS.
We ask the following question: is it true that the variables under study are in fact
influenced, not by log(n) itself, but by the excess number of representatives, defined as the
residual of the crude log-linear regression, log(n) log(bn) = log(n) (0.4) log(N) 4.32.
This hypothesis can be tested by means of the standard F -test of a single linear restiction,
because using log(n) log(bn) as a control instead of log(n) and log(N) is tantamount to
assuming that the coefficient on log(N) is equal to 0.4 times the coefficient on log(n).
According to some theories, we should observe more restrictions to trade, and thus
less openness, in countries with a smaller legislature: protectionist policies would be easier
to implement with a relatively smaller parliament.
19
But the impact of log(n) on trade
19
On the other hand, since there exists a positive correlation between an economy’s exposure to interna-
29
openness is not significant. Table 2 presents a study of the possible impact of log(n) or
(log(n) log(bn)) on F REEOP . Column (1) in Table 2 is the unconstrained version of the
regression, while the constrained version is given by column (3). However log(n) log(bn) is
not significant in regression (3). Column (2) yields good results, but the significant coefficient
on log(n) is likely to indirectly capture the effect of log(N), which has been removed from
the regression. Total population and land surface are significant with the expected negative
sign, as shown by column (4). This result is robust to the addition of many controls.
20
We
find more interesting results with the other indices.
TABLE 2
Dependent Variable: FREEOP
(1) (2) (3) (4)
Constant 0.09 0.22 0.11 0.14
(1.3) (3.8)*** (3.33)*** (5.39)***
log(n) 0.01 -0.024
(0.83) (-2.39)**
log(N) -0.02 -0.02
(-3.09)*** (-3.09)***
log(GNP) 0.015 0.017 0.02 0.02
(3.76)*** (4.65)*** (3.05)*** 4.48***
LAND -1.6×10
5
-1.89×10
5
-2.19×10
5
-1.65×10
5
(-5.3)*** (-5.31)*** (-5.5)*** (-5.37)***
DEM46 0.05 0.054 0.048 0.05
(2.51)** (2.53)*** (2.53)** (2.69)***
log(n)-log(bn) 0.01
(0.49)
No. Obs. 67 67 67 67
R
2
0.65 0.61 0.57 0.64
Adjusted R
2
0.62 0.58 0.54 0.62
Sum squared Resid 0.13 0.15 0.16 0.13
All columns were estimated by ordinary least squares. White heteroskedastic-consistent standard errors are
used to calculate t-statistics, which are reported in parentheses. Significance is denoted by *** (1%); **
(5%); * (10%).
The first dependent variable studied in Table 3 , ST AT EINT ERF , provides a mea-
tional trade and the size of the government (e.g. Rodrik (1998), Alesina and Wacziarg (1998)), FREEOP
could presumably be positively related to the number of representatives.
20
The result holds when we sequentially add: AFRICA, COMMONLAW, ELF, FEDERAL, FORMBRIT-
COL, PERCENTPROT, GOVWAGE. None of these variables have a coefficient significantly different from
zero.
30
sure of barriers to business for existing firms (i.e., whether State interference hinders the
development of business). According to some theories, we should observe larger barriers
to business in countries with a smaller legislature. The rent-seeking strategies of lobbies
would be easier to carry out with a smaller number of seats in parliament. Yet, in Table
3, column (1a) exhibits a positive and significant coefficient on log(n). This result is robust
to the addition of many controls.
21
It seems that the residual of the regression of log(n) on
log(N) is in fact the appropriate explanatory variable. Column (1a) is the unconstrained
version of the regression, and column (1b) is the constrained version, exhibiting a positive
and significant coefficient on (log(n) log(bn)). The F -test testing the model of column (2b)
against that of column (2a) yields (45 5)(13.32 12.32)/12.32 = 3.246. The critical value
being F
95
(1, 40) = 4.08 at the 5% level, we cannot reject the assumption that it is the excess
number of representatives, log(n) log(bn), which has an impact on ST AT EINT ERF .
The second variable, SUNKCOST , measures barriers to entry for entrepreneurs
(i.e., barriers to the creation of new firms). This variable has been shown to be a major
determinant of the size of a country’s informal sector, and also to contribute to the level of
rents in the legal sector (see Auriol and Warlters (2005), Ciccone and Papaioannou (2007)).
Again, we have reasons to expect higher barriers to entry in countries with relatively smaller
legislatures. Yet, column (2a) in Table 3 shows the opposite result: the coefficient on log(n)
is positive and significant at the 5% level. The result is again robust to the addition of many
controls.
22
It is also robust when the regression is run without France and Italy, which have
been identified as outliers above.
23
The unconstrained regression is given by column (2a),
while the constrained regression, with log(n) log(bn) as a regressor, is given by column
(2b). Note that log(n) log(bn) has a significant coefficient in column (2b). The F -test
comparing columns (2a) and (2b) is (71 5)(20.46 20.05)/20.05 = 1.35, and the critical
21
The result holds when we sequentially add, AFRICA, ELF, COMMONLAW, LAND, FEDERAL,
FORMBRITCOL, PERCENTPROT, GINI, GOVWAGE, OECD, TRANS, to the regression.None of theses
variables have a significant coefficient.
22
The result holds when we sequentially add to the regression:AFRICA, COMMONLAW, DEM46, LAND,
FEDERAL, FORMBRITCOL, GINI, GOVWAGE. None of these variables have a coefficient which is signif-
icantly different from zero.
23
The coefficient of the log-number of representatives is then positive and significant at the 1% level.
31
value is F
95
(1, 66) 4, so we cannot reject the assumption that it is the residual of the crude
log-linear regression which has an impact on SUNKCOST .
TABLE 3
Dependent Variable: STATEINTERF STATEINTERF SUNKCOST SUNKCOST
(1a) (1b) (2a) (2b)
Constant 1.58 4.65 0.56 2.32
(1.45) (6.22)*** (1.18) (3.38)***
log(n) 0.47 0.45
(2.2)** (2.26)**
log(N) -0.08 -0.25
(-0.63) (-2.29)**
log(GNP) -0.19 -0.26 -0.23 -0.23
(-2.31)** (-3.0)*** (-3.05)*** (-3.07)***
DEM46 -0.4 -0.38
(-2.27)** (-1.81)*
TRANS -0.34 -0.3
(-2.16)** (-2.07)**
log(n)-log(bn) 0.51 0.42
(2.08)** (2.22)**
No. Obs. 45 45 71 71
R
2
0.43 0.38 0.3 0.28
Adjusted R
2
0.37 0.34 0.25 0.25
Sum squared Resid 12.32 13.32 20.05 20.46
All columns were estimated by ordinary least squares. White heteroskedastic-consistent standard errors are
used to calculate t-statistics, which are reported in parentheses. Significance is denoted by *** (1%); **
(5%); * (10%).
The results of Table 3 suggest that it is the part of log(n) which is unexplained by
total population (i.e., the excess number of representatives) which is in fact causing a higher
level of barriers to entry. These results show a negative correlation between the number of
representatives and the degree of laissez-faire (or free-market orientation) of a country. We
are not aware of a theory explaining these facts, but they suggest a possible straightforward
“quantity theory” of the legislators’ activity and meddling in the operation of markets: the
more representatives, the more people work on law and regulation; the higher their “output”,
and the more they meddle in business activity. A related idea is provided by Becker (1983,
p 388):
“Cooperation among pressure groups is necessary to prevent the wasteful ex-
32
penditures on political pressure that result from the competition for influence.
Various laws and political rules may well be the result of cooperation to reduce
political expenditures, including restrictions on campaign contributions and the
outside earnings of Congressmen, the regulation of and monitoring of lobbying
organizations, and legislative and executive rules of thumb that anticipate (and
thereby reduce) the production of pressure by various groups.”
To this list, we add the number of representatives itself. It might well be that in some
countries, the low number of representatives is a long-established, endogenous response of
the political system, reflecting cooperation among various forces to reduce lobbying and
inefficient state interventions. If these ideas are true, there are some chances that we will
find a positive correlation between the number of representatives and corruption.
5.4 The Number of Representatives, and Corruption
We finally study the possible impact of the number of representatives on the level of perceived
corruption in a country. Treisman (2000) has analyzed the causes of perceived corruption in a
cross-national study, using T ISCORE, a well-known corruption index computed by Trans-
parency International. Treisman (2000) finds that countries with more developed economies,
protestant traditions, histories of British rule, long exposure to democracy and higher im-
ports are less “corrupt”. Federal states are more “corrupt”. We have reproduced Treisman’s
regressions with additional controls and chiefly, we added the number of representatives in
the regressions.
The results are presented in Table 4; they strongly suggest that the number of representatives
has a positive impact on corruption. Columns (1), (2), and (3) of Table 4 reproduce columns
(2), (3), and (4), respectively, in Treisman (2000, table 3, p 58), for the year 1996, with the
addition of the number of representatives n.
24
With these changes, contrary to Treisman’s
results, F ORMBRIT COL is not significant, while DEM46 becomes significant.
25
24
The dataset is not exactly the same: more countries are included in our results. Some variables are
different: we use F REEOP instead of the imports/GNP ratio in Treisman’s work. RAW M AT is the
percentage of raw materials in the country’s exports.
25
This is true, with or without the number of representatives.
33
TABLE 4
Dependent Variable: TISCORE
(1) (2) (3) (4)
Constant 14.94 13.69 13.82 12.17
(13.28)*** (11.31)*** (10.14)*** (10.21)***
n 0.0018 0.0017 0.002 0.0023
(2.61)** (2.07)** (1.93)* (3.51)***
COMLAW -0.18 -0.17 -0.12
(-0.45) (-0.46) (-0.32)
FORMBRITCOL -0.72 -0.56 -0.64
(-1.61) (-1.28) (-1.46)
PERCENTPROT -0.026 -0.014 -0.01 -0.01
(-3.85)*** (-2.15)** (-1.77)* (-3.22)***
ELF -0.24 -0.2 -0.42
(-0.33) (-0.31) (-0.57)
RAWMAT -0.008 -0.013 -0.01
(-0.69) (-1.19) (-1.16)
log(GNP) -1.29 -1.02 -1.03 -0.81
(-9.24)*** (-6.8)*** (-5.85)*** (-5.08)***
FEDERAL 0.95 0.93
(2.35)** (2.06)**
DEM46 -1.53 -1.72 -1.63
(-3.34)*** (-3.57)*** (-4.73)***
FREEOP 0.3
(0.13)
AFRICA -1.12
(-2.81)***
OECD -0.76
(-1.95)*
DENSITY -0.0005
(-4.73)***
No. Obs. 69 69 56 79
R
2
0.70 0.75 0.76 0.77
Adjusted R
2
0.67 0.71 0.71 0.75
Sum squared Resid 115.03 96.71 80.65 101.74
Columns (1) to (4) were estimated by ordinary least squares. White heteroskedastic-
consistent standard errors are used to calculate t-statistics, which are reported in
parentheses. Significance is denoted by *** (1%); ** (5%); * (10%).
But the novelty is of course that the coefficient on n is positive and significant in the 4
variants presented in Table 4. The result is thus fairly robust.
26
We are tempted to interpret
26
The significant coefficient on n in column (4) is robust to the inclusion of many controls, i.e., ELF,
COMLAW, LAND, FEDERAL, FORMBRITCOL, GINI, GOVWAGE, TRANS. None of these variables
have significant coefficients.
34
this fact as a confirmation of the “quantity theory” sketched above. More representatives
produce more red tape and induce more corruption.
To sum up, putting together the results of Tables 3 and 4, suggests that the number
of representatives really matters. Political regimes in which the rate of representation is low,
the influence and “value” of each representative are correlatively high, could paradoxically
be regimes in which the supply of intervention is less elastic, and the occasions for corruption
more limited. These results are of course just an indication that the subject deserves more
attention. Additional work is needed to check for robustness and causality.
6 Conclusion
We have proposed a model of a representative democracy, based on a two-stage model of
constitution design, with a constitutional and a legislative stage. This model embodies
a notion of political stability of the constitution, called robustness, which emphasizes the
idea that the founding fathers do not know the distribution of citizens’ preferences. From
this model, we derived a “square-root formula” for the number of representatives, stating
that the optimal number should be proportional to the square root of total population.
Regression work on a sample of more than a 100 countries shows that the number of national
representatives is proportional to total population to the power of 0.4 : the square-root
theory is almost true. We then find that the USA is an outlier with too few representatives,
while France and Italy, for instance, have too many. The quality of fit is lower when data
on the 50 US state legislatures is used. We finally cannot reject the assumption that the
excess number of representatives has an impact on the degree of state interference and on
an index of barriers to entry of new firms (i.e., red tape). The number of representatives
itself has a significant, positive impact on the degree of perceived corruption. The number
of representatives thus matters and we suggest that a “quantity theory” of representatives
hold: more seats in parliament are associated with more red tape, more state interference in
business, and more corruption.
35
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Tocqueville, Alexis de (1835), De la emocratie en Am´erique, vol. 1., reprinted 1981,
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Treisman, Daniel (2000), “The Causes of Corruption: A Cross-National Study”, Journal of
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39
8 Appendix
The formal statement of the result used in the derivation of the theorem is as follows.
Proposition A1. Assume that there are at least three possible decisions, that the separation-
of-powers, subsidiarity and anonymity principles hold. Assume that any utility function v
is possible ( V is a universal domain). Then, (f, t) is nonmanipulable if and only if the
following three conditions hold:
f(bv) arg max
qQ
(
k(q) +
n
X
i=1
bv
i
(q)
)
(27)
where k is an arbitrary fixed function, and for all i = 1, ..., n;
t
i
(bv) =
X
j6=i
bv
i
(f(bv)) k(f(bv)) + m(bv
i
), (28)
where m is an arbitrary fixed function that doesn’t depend on v
i
; and finally,
k, and m are fixed in the constitution. (29)
See Auriol and Gary-Bobo (2007) for a proof of this result, which is an adaptation of the
classic characterization of dominant strategy mechanisms, under the assumption of quasi-
linear preferences. The hard part in the proof of this proposition is the “only if” part;
it heavily relies on K. Roberts’ (1979) characterization theorem. It is intuitive that the
requirement of dominant strategies restricts the set of admissible mechanisms in such a way,
even if it is difficult to prove that these mechanisms are the only nonmanipulable ones. Note
that these mechanisms are also budget-balanced by construction, because there is at least
one citizen which is not a representative (i.e., at least agent 0 does not report about his
(her) preferences). We further assume that the FF are constrained to choose f(.) in the set
defined by Proposition A1 above, even if the domain V is restricted to a particular class of
utility functions (to keep matters simple.
27
) We now provide a short proof of the Lemmas.
27
In the case of quadratic preferences, it is well-known that there exists a fully optimal, budget-balanced,
Groves mechanism: but it is a member of the same family (see Moulin (1988), chapter 8, Groves and Loeb
(1975)). In the quadratic case, we can design the transfers so as to “isolate” the representatives: they can
self-finance their revelation incentives.
40
Proof of Lemma 1. Let
P
N
i=0
θ
i
= r + s,where r =
P
n
i=1
θ
i
and s = θ
0
+
P
N
i=n+1
θ
i
. Then,
W
P
(a, b) = E
P
(r + s)
r + ab
n + b
(N + 1)(r + ab)
2
2(n + b)
2
nF.
Using the fact that E
P
(r) =
P
, E
P
(s) = (N + 1 n)µ
P
, E
P
(rs) = E
P
(r)E
P
(s) (because
of independence), E
P
(r
2
) =
2
P
+ n
2
µ
2
P
and after some elementary computations, we find
W
P
(a, b) =
1
N + 1
2(n + b)
2
P
n + b
+
n(N + 1)
(n + b)
2
n
2
+ b
µ
2
P
+
ab
2
(N + 1)
(n + b)
2
µ
P
a
2
b
2
(N + 1)
2(n + b)
2
nF. (30)
We then take the expectation of W
P
(a, b) with respect the the prior distribution B. This
yields the stated result.
Q.E.D.
Proof of Lemma 2. We first maximize the expression of E
B
[W
P
(α, β)] given by Lemma 1
with respect to a. This is equivalent to maximizing (αbµ α
2
/2). Hence, α
= bµ. Substitute
next α
= bµ in W (α, β) E
B
[W
P
(α, β)]. We then easily obtain
E
B
[W
P
(bµ, β)] =
n + β
N + 1
2
nbσ
2
(n + β)
2
+
n(N + 1)
(n + β)
2
n
2
+ β
(bµ
2
+ bz
2
)
+
bµ
2
β
2
(N + 1)
2(n + β)
2
nF. (31)
We finally maximize W (bµ, β) with respect to β. After some simplifications, we find the
first-order condition
W (bµ, β)
β
=
n
(n + β)
3
bσ
2
(N + 1 n β) bz
2
β(N + 1)
= 0.
We then solve this equation for β
. It is easy to check that W is strictly quasi-concave and
it follows that β
is the unique global maximizer of W (bµ, β).
Q.E.D.
41
TABLE A1. Dependent Variable: log
n
(1) (2) (3) (4) (5)
Constant 3.4*** 4.21*** 4.03*** 4.22*** 4.02***
(7.16) (9.97) (9.34) (10.21) (11.67)
log(N) 0.45*** 0.41*** 0.4*** 0.4*** 0.39***
(15.27) (16.79) (16.66) (16.16) (18.32)
log(GNP) 0.08** 0.07** 0.06* 0.04 0.02
(2.11) (2.35) (1.69) (1.09) (0.76)
log(TAXREV) 0.3*** 0.16* 0.12 0.16* 0.15*
(3.19) (1.94) (1.37) (1.94) (1.91)
DENSITY -0.0001** -0.0001*** -0.0001*** - 0.0001*** - 0.0001***
(-2.27) (-3.44) (-2.82) (-2.67) (-2.9)
GINI -1.63*** -2.1*** -1.37** -1.57*** -0.71**
(-3.49) (-4.32) (-2.24) (-2.72) (-2.21)
ELF -1.22*** -1.31*** -0.83* -0.98**
(-2.67) (-3.15) (-1.74) (-2.07)
GINI×ELF 2.7*** 2.89*** 1.69 2.18**
(2.74) (3.26) (1.6) (2.16)
DEM46 -0.19 -0.21* -0.22* -0.24** -0.2*
(-1.27) (-1.92) (-1.82) (-2.31) (-1.96)
PERCENTPROT -0.0006
(-0.34)
FEDERAL 0.009
(0.085)
COMLAW -0.13
(-1.1)
FORMBRITCOL 0.06
(0.54)
OECD 0.24** 0.26** 0.31***
(2.08) (2.13) (3.11)
TRANS 0.10
(0.85)
AFRICA 0.14
(1.28)
No. Obs. 71 93 93 93 111
R
2
0.88 0.86 0.87 0.86 0.83
Adjusted R
2
0.86 0.84 0.85 0.85 0.82
Sum squared Resid 4.51 7.4 6.9 7.06 9.5
Columns (1)–(5) were estimated by ordinary least squares. White heteroskedastic-consistent standard errors
are used to calculate t-statistics, which are reported in parentheses. Significance is denoted by *** (1%); **
(5%); * (10%). PERCENTPROT is the percentage of protestants in the country; OECD indicates an OECD
country, TRANS a former socialist (i.e, transition) country; FEDERAL a country with a federal structure
(e.g., the US, Germany); AFRICA indicates a country form the African continent; FORMBRITCOL indicates
a former British colony and the UK; COMLAW indicates a country with a Common Law system; DEM46
equals 1 if the country has been democratic in all 46 years (1950-1995) (source Treisman (2000)).
42
Table A2: Actual and fitted number of representatives
Value bn has been computed based on regression 4 in Table 1. For missing data, reported in
italic, regression 1 in Table 1 has been used.
COUNTRY n bn
Albania 140 113.21610
Angola 220 187.31671
Argentina 329 285.41909
Armenia 190 129.68857
Australia 219 308.46221
Austria 247 293.03681
Azerbaijan 350 170.93651
Bangladesh 300 460.99839
Belgium 221 262.90633
Benin 83 121.36892
Bolivia 157 140.49486
Bosnie-Herz 240 128.93847
Brazil 594 463.96416
Bulgaria 240 213.43878
Burkina-Faso 227 155.45687
Cambodia 120 144.75544
Cameroon 180 161.83940
Canada 399 336.39994
Cl Africa Rep. 85 79.448245
Chile 167 178.12343
Colombia 267 233.24201
Costa Rica 57 114.81261
Coast Ivory 175 188.57065
Croatia 201 143.04017
Czech Rep. 281 251.82664
Denmark 175 243.65874
Dominican Rep. 150 138.41592
Egypt 664 460.33356
El Salvador 84 113.00308
Equ. Guinea 80 39.270870
Estonia 101 89.256839
Fiji 104 63.739114
Finland 200 218.41965
France 898 545.84014
Gabon 120 80.252787
Germany 740 661.84103
Ghana 200 182.14392
Greece 300 231.76985
Grenada 28 24.882613
43
COUNTRY n bn
Guatemala 116 142.01617
Guyana 65 69.559220
Honduras 128 117.29668
Hungary 386 277.03339
Iceland 63 54.259966
India 790 860.18700
Indonesia 500 542.36577
Ireland 226 165.18585
Israel 120 175.02468
Italy 945 570.24767
Jamaica 81 103.79495
Japan 763 704.17869
Jordan 120 141.44161
Kazakhstan 177 238.86700
Kenya 188 239.66050
Rep. of Korea 299 415.62166
Kyrgyr. Rep. 105 139.26255
Latvia 100 110.94319
Lebanon 128 84.179405
Lesotho 65 82.219469
Lithuania 141 129.08048
Macedonia 120 98.900478
Madagascar 138 157.81119
Malawi 177 155.33208
Malaysia 212 239.34138
Mali 129 145.56781
Mauritania 135 90.158273
Mauritus 62 67.472499
Mexico 628 389.86490
Moldava 104 137.72470
Mongolia 76 112.42436
Mozambiq. 250 165.65331
Nanibia 72 91.836596
Nepal 255 198.00898
Netherlands 225 325.95926
New Zealand 99 152.13716
Nicaragua 92 116.27401
Niger 83 127.39640
Norway 165 219.39632
Pakistan 304 448.11270
Panama 72 89.919477
Papua Guin. 109 121.11107
Paraguay 125 105.57714
44
COUNTRY n bn
Peru 120 239.17824
Phlippines 250 351.30956
P